Ok, so I know you just clicked on this blog title and thought "Wow, Julia, she's really getting to the point. I mean, maybe I will just stop singing!" No, no, no. This is not what it's about. . . That requires a lot more serious of a discussion, and I will give you my short answer to that question: No.
This is about singing with the mouth closed or shut. There is so much hot debate on both sides of the coin. Some technicians say the mouth MUST be open at all times. Other pedagogues argue that a more closed mouth position optimizes resonance and laryngeal freedom. I would like to be Mr. Benedict Arnold and take both sides for a moment.
Check out this drawing above. Do you see how the tongue attaches to the mandible (the jaw)? Can you see how intimately all these structures are related? Now imagine having a fixed jaw position in either direction. Can you feel the tension that slowly creeps in as you try to maintain a "fixed" jaw position? It has been proven that muscles fatigue more quickly in a static state rather than in movement. Thus, holding our jaw in any one position is never a good idea. But lets dive a little further. . .
Why might a pedagogue advocate an open mouth posture? For several reasons actually. Most singers come with imbalances, and one of the most common issues is with the mobility of the TMJ (temporomandibular joint- or jaw joint). A singer "should" be able to open and close their mouth loosely, switfly, and without discomfort. However, because of tension held in the masseter, temporalis, and other jaw muscles, it can be difficult for a singer's jaw to release. In this case, exercising the opening action of the jaw can help strengthen the jaw opening muslces (anterior digastric, mylohyoid, and inferior head of the lateral pterygoid). Naturally, our jaw closing muscles are much stronger (masseter, temporalis, medial pterygoid, and superior head of the lateral pterygoid), and that is not a good situation if we cannot open our jaw freely. Thus, I do advocate vocalizing frequently with an open jaw posture. However, this should not be retained dogmatically when singing.
So, now, let me play the other side of the fence. A closed mouth posture can often help the tongue find a higher position in the mouth, especially if there is some tongue tension. A higher tongue position prevents the tongue from "sitting" on the hyoid bone (the little floating bone that sits above the larynx), thus reducing laryngeal freedom and vocal ring and range. I recommend vocalizing in the beginning of a session with a closed mouth position on a hum and then a closed mouth "ziiiii", This allows the tongue to find its natural placement without having to work too hard maintaining its position with the jaw open.
You may subscribe to either side of the camps, but I like to stay in the middle. Sing where the line and words take you. Some notes will be more open, some will be more closed. The jaw should not be "placed" or "locked" into position. Play around and find freedom.